Pi^3 == (1/16) Sum[((-1)^k/1024^k) (32/(1 + 4 k)^3 + 8/(2 + 4 k)^3 + 1/(3 + 4 k)^3), {k, 0, Infinity}] + (5/2) Sum[((-1)^k/64^k) (32/(1 + 12 k)^3 - 192/(2 + 12 k)^3 + 88/(3 + 12 k)^3 - 8/(5 + 12 k)^3 + 84/(6 + 12 k)^3 - 4/(7 + 12 k)^3 + 11/(9 + 12 k)^3 - 12/(10 + 12 k)^3 + 1/(11 + 12 k)^3), {k, 0, Infinity}]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msup> <mi> &#960; </mi> <mn> 3 </mn> </msup> <mo> &#63449; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 16 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <msup> <mn> 1024 </mn> <mi> k </mi> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 8 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 32 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <msup> <mn> 64 </mn> <mi> k </mi> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 192 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> </mrow> <mo> + </mo> <mfrac> <mn> 88 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <mn> 8 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 84 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <mn> 4 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 11 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <mn> 12 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 10 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 11 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 32 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <power /> <pi /> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 16 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 1024 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 5 <sep /> 2 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 64 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 192 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 88 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 5 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 84 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 7 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 11 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 9 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 10 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 11 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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G.Huvent (2006)

2007-05-02